Definition of Limit

Q. the value of $\sum _{m=1}^{10}{I}_{2m+1}$ is equal to

1. $0$
2. $5\pi$
3. $10\pi$
4. None of these

Q. If ${\int }_a^b{x}^{3}dx=0$ and ${\int }_a^b{x}^{2}dx=\frac{2}{3}$, then what are the values of $a$ and $b$ respectively?

1. $-1,1$
2. $1,1$
3. $0,0$
4. $2,-2$

Q. If $f\left(y\right)=e^y,g\left(y\right)=y,y>0$ and $F\left(t\right)={\int }_0^{t}f(t-y)g\left(y\right)dt,$ then

1. $F\left(t\right)=e^t-(1+t)$
2. $F\left(t\right)=t{e}^t$
3. $F\left(t\right)=t{e}^{-t}$
4. $F\left(t\right)=1-e^t(1+t)$

Q. Statement ${\int }_{-a}^{a}f\left(x\right)dx=\{\begin{array}{cc}2{\int }_0^{a}f\left(x\right)dx,& \text{if f is an even function}\\ 0& \text{if f is an odd function}\end{array}$ is ?

1. True
2. False

Q. If ${\int }_0^{\pi }\frac{x^2\cos x}{(1+\sin x{)}^2}dx=A\pi -{\pi }^2$ then A is

Q. If ${\int }_0^{\pi }\ln \sin xdx=k$, then the value of ${\int }_0^{\pi /4}\ln (1+\tan x)dx$ is:

1. $-\frac{k}{4}$
2. $\frac{k}{4}$
3. $-\frac{k}{8}$
4. None of these
   

Q. ${\int }_0^1{\tan }^{-1}\left[\frac{2x-1}{1+x-x^2}\right]dx=$

1. 0
2. 1/2
3. 1
4. $\pi /6$

Q. Evaluate ${\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+\cos x+\sin x}dx$

Q. If ${\int }_0^{\pi }xf\left(\sin x\right)dx=A{\int }_0^{\pi /2}f\left(\sin x\right)dx,$ then $A$ is equal to:

1. $0$
2. $\pi$
3. $\frac{\pi }{4}$
4. $2\pi$

Q. ${\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}dx=\frac{\pi }{4}$

1. True
2. False